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Machine learning models have emerged as powerful tools in physics andengineering. Although flexible, a fundamental challenge remains on how toconnect new machine learning models with known physics. In this work, wepresent an autoencoder with latent space penalization, which discovers finitedimensional manifolds underlying the partial differential equations of physics.We test this method on the Kuramoto-Sivashinsky (K-S), Korteweg-de Vries (KdV),and damped KdV equations. We show that the resulting optimal latent space ofthe K-S equation is consistent with the dimension of the inertial manifold. Theresults for the KdV equation imply that there is no reduced latent space, whichis consistent with the truly infinite dimensional dynamics of the KdV equation.In the case of the damped KdV equation, we find that the number of activedimensions decreases with increasing damping coefficient. We then uncover anonlinear basis representing the manifold of the latent space for the K-Sequation.

Learning a Reduced Basis of Dynamical Systems using an Autoencoder